Police allege the Audi lost control while driving along Ridge Road, Kalorama, before crashing through a stone barrier and into the front of the art gallery around 3am on Saturday.
The six teenagers had fled the scene by the time police arrived but officers found and arrested a teenage girl nearby.
While police were searching the area, they received a report of an attempted car theft nearby.
Police arrived at Price Road, the location of the attempted car theft, and there they noticed a ride-share car.
“They flagged the driver down and located four teenage passengers, three males and one female, who were all taken into custody,” Victoria Police said in a statement.
“The Dog Squad rounded up the last male youth.”
Five teens are being interviewed by police while the sixth has been taken to hospital with a leg injury after being bitten by the police dog.
The art gallery suffered extensive damage during the crash.
